Nielsen moved in on goalie Robin Lehner and put a backhand in on the Islanders’ second and last shot of the shootout.
The Islanders (5-5-3) outshot the Senators 57-31 through overtime and got goals from Matt Martin, John Tavares and Pierre-Marc Bouchard in regulation. Evgeni Nabokov made 27 saves for New York, which rallied from a pair of two-goal deficits.
Erik Condra, Bobby Ryan, Mika Zibanejad and Clarke MacArthur scored for the Senators (4-6-3).
Lehner made the biggest contribution with 53 saves, including one against Thomas Vanek on a breakaway in overtime.
Ottawa took a 4-3 lead into the second intermission after a frantic finish to the period, in which two Senators were in the penalty box for the final 55 seconds.
At the start of the third, the Islanders still had 57 seconds left on their 5-on-3 power play, and they had clean ice to skate on, but they were unable to score.
However, New York got even 4-4 at 3:14 on Nielsen’s goal.
There were a total of five goals in the second period as the Islanders scored twice in 28 seconds to tie the game 2-2 before the Senators rebounded to grab a 4-2 lead.
Tavares had an empty net to deposit the puck into after a blind pass from Vanek found him alone in front. At 9:35 of the period, Martin tied it when he tipped a point shot from former Senators player Matt Carkner past Lehner.
The Senators responded with goals from Ryan and MacArthur at 12:34 and 14:47.
With the teams playing 4 on 4 following a pair of minor penalties, Bouchard put a rebound past Lehner from a sharp angle to pull New York back to within a goal.
Zack Smith was the Ottawa player in the box at the time. Thirty seconds after Bouchard’s goal, Smith emerged from the penalty box but barely made it 10 feet before being penalized for interference 3 seconds after his first infraction ended.
Just 7 seconds later, Jared Cowen was called for holding, setting up the Islanders’ 5-on-3 advantage.
Ottawa took a 2-0 lead in the first on goals by Condra, with his first of the season, and Zibanejad. Condra left the game in the second period because of a muscle strain in his leg and didn’t return.
NOTES: Derek Grant’s first-period assist was his first NHL point in 11 career games, all with the Senators. ... Vanek played in his 600th NHL game and second with the New York Islanders. He earned his first point with his new team after recording 497 in 598 games with Buffalo. ... Matt Kassian and Joe Corvo were scratched for the Senators. Colin McDonald, Eric Boulton and Brian Strait didn’t dress for the Islanders.